A small spehre of mass `m` is connected by a stirng to a nail at `O` and moves in a circle of radius `r` on the smooth plane inclined at an angle ` theta` with the horizontal. If the sphere has a velocity `u` at the top position A. Mark the correct options.

Options

A. Minimum velocity at `A` so that the string does not get slack instanntaneously is `sqrt(3/5gr)`
B. Tension at `B` is `8mg//5` if the sphere has the required velocity as in option a.
C. Tension `C` is `18mg//5` in situation of option b.
D. none of these

Answer

A, C, D

Explanation

Minimum velocity at A should be
`v_A=sqrt(gsin37^@r)=sqrt(3/5gr)`
Velocity at `B:v_B^2=v_A^2+2g r sin 37^@`
`implies v_B^2=3/5gr+6/5grimpliesv_B=sqrt((9gr)/5)`
`T_B=(mv_B^2)/r=(9mg)/5`
Velocity at `C:v_C^2 =v_A^2+2g2rsin37^@`
`implies v_C^2=3/5gr+12/5grimpliesv_C=sqrt(3gr)`
`T_C-mg sin 37^@=(mv_C^2)/r`
`implies T_C-3/5mg=3mgimpliesT_C=18/5mg`